This document discusses personnel scheduling problems and various solution approaches using constraint programming techniques. It begins by describing the multi-activity shift scheduling problem and typical constraints such as minimum and maximum shift lengths and break requirements. It then discusses using regular language constraints and context-free grammar constraints to model the sequencing rules and filter valid schedules. Decomposing global constraints and using table or grammar graph constraints are described as alternatives to custom propagators.

1. One
Problem,
Two
Structures,
Six
Solvers,
and
Ten
Years
of
Personnel
Scheduling
1
!
Louis-­‐Mar@n
Rousseau
CIRRELT
-­‐
École
Polytechnique
de
Montréal
!

2. ShiI
Scheduling:
Once
upon
a
@me...
• It
all
started
in
1950’s...
2

3. Personnel
scheduling:
nowadays
Clopening happens
when one works
both a closing and
the following
opening shift
3

4. Outline
• GOAL:
illustrate
that
the
right
structures
for
a
problem
can
be
useful
whatever
the
solu:on
technology
!
• Using
formal
languages
structures
in:
– CP,
SAT
and
MIP
– Hybrid
CP-­‐MIP
Branch
and
Bound
– Dynamic
Programming
• with
Large
Neighborhood
Search
• with
Column
Genera@on
!
• Discuss
two
applica@ons
using
CP
techniques
4

5. !
The
mul@-­‐ac@vity
ShiI
Scheduling
Problem
5
A very extensive surrey of the mono-activity version in:
!
• Ernst, A. T., Jiang, H., Krishnamoorthy, M., & Sier, D. (2004). Staff
scheduling and rostering: A review of applications, methods and models.
European journal of operational research, 153(1), 3-27.

6. Problem
descrip@on
• Ac@vi@es
(what
employees
can
do)
• Time
Periods
(15
minutes)
• ShiIs
• Star@ng
period
• End
period
• Work
segments
• Rest
segments
!
• Assign
Rest activities:
shiIs
to
employees
that
meet
– work
regula@ons
– their
exper@se
– their
availabili@es
!
• Op@mize
– Labor
costs
– divergence
from
desired
number
of
employees
Work activities:
Shift 1:
Shift 2:
6
5
4
3
1
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

7. A
typical
example
1. If
an
employee
is
working,
he
must
work
between
3
hours
and
8
hours
2. If
a
working
employee
covers
at
least
6
hours
of
work
ac@vi@es,
he
must
have
two
15
minute-­‐breaks
and
a
lunch
break
of
1
hour.
3. If
a
working
employee
covers
less
than
6
hours
of
work
ac@vi@es,
he
must
have
a
15
minute
break,
but
no
lunch.
4. If
performed,
the
dura@on
of
any
ac@vity
a
∈
W
is
at
least
1
hour
5. A
break
(or
lunch)
is
necessary
between
two
different
work
ac@vi@es.
6. Work
ac@vi@es
must
be
inserted
between
breaks,
lunch
and
rest
stretches.
7. Rest
shiIs
have
to
be
assigned
at
the
beginning
and
at
the
end
of
the
day.
8. Ac@vi@es
a
∈
Ft
are
not
allowed
to
be
performed
at
period
t
∈
T.
Minimize labor costs as well as over and under covering costs

8. !
Constraints
based
on
Regular
&
Context-­‐free
Grammar
languages
8
• Pesant, G. (2004). A regular language membership constraint for finite sequences of variables. In Principles and
Practice of Constraint Programming–CP 2004 (pp. 482-495). Springer Berlin Heidelberg.
• Beldiceanu, N., Carlsson, M., & Petit, T. (2004). Deriving filtering algorithms from constraint checkers. In Principles
and Practice of Constraint Programming–CP 2004 (pp. 107-122). Springer Berlin Heidelberg.
• Sellmann, M. (2006). The theory of grammar constraints. In Principles and Practice of Constraint Programming-CP
2006 (pp. 530-544). Springer Berlin Heidelberg.
• Quimper, C. G., & Walsh, T. (2006). Global grammar constraints. In Principles and Practice of Constraint
Programming-CP 2006 (pp. 751-755). Springer Berlin Heidelberg.
• Quimper, C.G., & Walsh, T. (2007). Decomposing global grammar constraints. In Principles and Practice of
Constraint Programming–CP 2007 (pp. 590-604). Springer Berlin Heidelberg.
• Kadioglu, S., & Sellmann, M. (2010). Grammar constraints. Constraints, 15(1), 117-144.
• Katsirelos, G., Narodytska, N., Walsh, T., (2011), The weighted grammar constraint, Annals of OR, 184:1,179-207
• Gange, G., Stuckey, P. J., & Van Hentenryck, P. (2013). Explaining Propagators for Edge-Valued Decision
Diagrams. In Principles and Practice of Constraint Programming (pp. 340-355).

9. A
regular
language
constraint
• Regular
language
membership
constraint
(CP)
– X
=
<x1,...,
xn>,
be
a
finite
sequence
of
variables
– Π
denote
a
DFA
or
NFA
– L(Π)
be
the
regular
language
recognized
by
Π
– aabbaa
and
ccc
are
words
of
L(Π),
acab
is
not.
9

10. Building
the
correct
DFA
(or
NFA)
Sequence
regula@ons Sequence
length
Λ

11. Filtering
Regular
in
CP
filtered domains!
X1 ∈ {a,b,c,o}!
X2 ∈ {o}!
X3 ∈ {a,c,o}!
X4 ∈ {a,o}!
X5 ∈ {a}
11
a
a
A simple automaton for rostering
1
2 5
b
a
b
c
o
3 6
b
o
b
4 7
c
o
c
c
a
o
a
o
o
o
a
xxxxx2 3 4 5
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
4
4
4
4
4
5
5
5
5
5
6
6
6
6
6
7
7
7
7
7
NNNNN2 5
2 5
1
1
2
2
3
3
1 x2 x3 x4 x5
1
1
2
2
3
3
4
4
1 1
4
4
2
2
5
5
3
3
6
6
4
4
7
5
7
5
6
6
7
7
NNNNNN1
2
3
4
5
6
7
1
2
3
4
5
6
7
1
2
3
4
5
6
7
1
2
3
4
5
6
7
1
b
b
c
o
3 6
b
o
b
4 7
c
o
c
c
a
o
a
o
o
o
a
1
2
3
4
5
6
7
1
2
3
4
5
6
7
1
2
3
4
5
6
7
1
2
3
4
5
6
7
1
2
3
4
5
6
7
x1 x2 x3 x4 x5
1
2
3
4
5
6
7
N1 N2 N3 N4 N5 N6
x1 x2 x3 x4 x5
1 1
1 1
2
2
3
3
4
4
5
5
6
6
7
7
2
2
3
3
4
4
5
5
6
6
7
7
1
1
2
2
3
3
4
4
5
5
6
6
7
7
1
1
2
2
3
3
4
4
5
5
6
6
7
7
5
5
6
6
7
7
N1 N2 N3 N4 N5 N6
Off
a
b
c
initial domains!
X1 ∈ {a,b,c,o}!
X2 ∈ {b,o}!
X3 ∈ {a,c,o}!
X4 ∈ {a,b,o}!
X5 ∈ {a}!
1
b
b
c
o
3 6
b
o
b
4 7
c
o
c
c
a
o
a
o
o
o
a
1
2
3
4
5
6
7
1
2
3
4
5
6
7
1
2
3
4
5
6
7
1
2
3
4
5
6
7
1
2
3
4
5
6
7
x1 x2 x3 x4 x5
1
2
3
4
5
6
7
N1 N2 N3 N4 N5 N6
x1 x2 x3 x4 x5
N1 N2 N3 N4 N5 N6
Forward pass Backward Pass

12. Decomposing
the
Regular
Constraint
• It
is
not
mandatory
to
introduce
a
new
propagator,
we
can
simply
introduce
new
variables
12
1
2 5
b
a
b
c
o
3 6
b
o
b
4 7
c
o
c
c
a
o
a
o
o
o
a
1
2
3
4
5
6
7
1
2
3
4
5
6
7
1
TableConstraint(Si, Xi, Si+1, Π), i : 1..5
a
2 5
Domain consistency of the table constraint will filter S such that!
S1 ∈ {1}!
∈ 1
S2 {1,2,3,4}!
2
S3 ∈ {1,5,6,7}!
3
S4 ∈ {1,2,4,5,7}!
4
S5 ∈ 5
{1,2,5,7}!
S6 ∈ 6
{2}
7
1
2
3
4
5
6
7
x3 x4 x5
N3 N4 N5 N6
Let Si be the state of the system at transition i:
Initially S1 ∈ {1} S{2..6} ∈ {1,2,3,4,5,6,7}
Every transition in the DFA is given !
as a tuple t = < OriginState, Value, DestinationState >
x1 x2 x3 x4 x5
1 1
2
3
4
5
6
7
2
3
4
5
6
7
1
2
3
4
5
6
7
1
2
3
4
5
6
7
1
2
3
4
5
6
7
1
2
3
4
5
6
7
N1 N2 N3 N4 N5 N6
b
b
c
o
3 6
b
o
b
4 7
c
o
c
c
a
o
a
o
o
o
a
1
2
3
4
5
6
7
1
2
3
4
5
6
7
1
2
3
4
5
6
7
1
2
3
4
5
6
7
1
2
3
4
5
6
7
x1 x2 x3 x4 x5
1
2
3
4
5
6
7
N1 N2 N3 N4 N5 N6
x1 x2 x3 x4 x5
1 1
2
3
4
5
6
7
2
3
4
5
6
7
1
2
3
4
5
6
7
1
2
3
4
5
6
7
1
2
3
4
5
6
7
1
2
3
4
5
6
7
N1 N2 N3 N4 N5 N6

13. On
our
example
1. If
an
employee
is
working,
he
must
work
between
3
hours
and
8
hours
2. If
a
working
employee
covers
at
least
6
hours
of
work
ac@vi@es,
he
must
have
two
15
minute-­‐breaks
and
a
lunch
break
of
1
hour.
3. If
a
working
employee
covers
less
than
6
hours
of
work
ac@vi@es,
he
must
have
a
15
minute
break,
but
no
lunch.
4. If
performed,
the
dura@on
of
any
ac@vity
a
∈
W
is
at
least
1
hour
5. A
break
(or
lunch)
is
necessary
between
two
different
work
ac@vi@es.
6. Work
ac@vi@es
must
be
inserted
between
breaks,
lunch
and
rest
stretches.
7. Rest
shiIs
have
to
be
assigned
at
the
beginning
and
at
the
end
of
the
day.
8. Ac@vi@es
a
∈
Ft
are
not
allowed
to
be
performed
at
period
t
∈
T.
13

14. A
Regular
language
for
our
example
Replacing
sequencing
rules
(4
to
8)

15. The
Grammar
Global
Constraint
• A
context
free
grammar
is
defined
by
– Σ
:
alphabet
or
ac@vi@es
(also
called
terminals)
– N:
a
set
of
characters
or
work
structures
called
non
terminals
– S
∈
N:
the
star@ng
character
– P:
a
set
of
produc@ons
in
the
form
A
-­‐>
w
where:
• A
is
a
non
terminal
• w
is
a
sequence
of
terminal
and
non
terminal
!
• Example:
!
!
A
sequence
of
length
3
can
only
be
:
aab
or
abb

16. CYK
Algorithm
S AB
A AA
B BB
A a
B b
FILTERING
S
B
A S B A S
B
A B A B A B
a b a b a b
16
A
dom(Xi)

17. The
Grammar
Graph
S
A B
A A B
B
a a b 17
b
• The
grammar
graph
(Γ)
encapsulates
all
feasible
solutions
• A
tree
in
Γ
represents
one
solu:on
Solu@on
aab
Solu@on
abb
(Γ) is a DNNF
(Γ) is hypergraph
and a solution (tree)
is an hyperpath

18. SbCPc
Sc , 2,5,7}
requirements about the length of the sequences they represent. Similarly, a production be restrained to be effective only at particular times of the day. We denote a literal Modeling
SSP
with
a
Grammar
Constraint
where P ⊆ N is the set of positions where A can occur in the and • We
A with APS
can
enrich
the
grammar
descrip@on
to
include
in
the
descrip@on
of
a
w→
S ⊆ N is the set of valid spans for the subsequence A. If the symbol omitted, produc@on
we assume that the set of valid positions and valid spans is N. A weight also • Valid
be set
associated of
periods
with (P)
a production. Each time a production is used in the an • Valid
employee, interval
for
the the
weight span
of
of sub
the sequence
production (S)
is added to the weight of the parsing indicate • Cost
of
the choosing
weight a
producw of @a on
production on the arrow as follows: A BC. One mention the weight whenever it is null.
For instance, the production A[2,4]
!
!
!
• These
8.9 → BC{2,5,7}
can be applied only if the [1,3] obtained has a length between 2 and 4 characters, the span of the literal C is between 3, and the can
subsequence be
easily
incorporated
C appears in
at the
positions CYK
algorithm.
2, 5, or 7. The weight associated production is 8.9 and the literal B is unconstrained.
These new restrictions on the productions are useful to model situations where must work a certain amount of time before changing activities or can have a only at lunch time. Weights on production can be used to encode preferences the case when extra salary must be given to an employee doing some over-time.
Implementing these constraints only require minor modifications to the algorithm. line 16 in Algorithm 1, when processing the production APa
• Here
is
the
enriched
grammar
that
captures
all
the
rules
(1..8)
18
7. An employee must have a break or a lunch before changing activities.
8. Periods of rest are to be scheduled at the beginning or the end of the day.
9. A part-time employee must work a minimum of 3 hours and fewer than 6 hours a day.
10. A full-time employee must work a minimum of 6 hours and a maximum of 8 hours a
day.
11. At specific times of the day, when the business is closed, every employee must either
rest, lunch, or be on a break.
As an example, a full time employee could rest between midnight and 9am, then work
on activity 1 from 9am to 10:30am, break, work on activity 2 from 10:45am to 12:00pm,
lunch, work on activity 3 from 1:00pm to 2:30pm, break, work on activity 4 from 2:45pm
to 5:00pm, and rest until midnight.
We model the rules using the following context-free grammar. Identifiers between de-limiters
⟨. . .⟩ are terminal symbols.
R → ⟨rest⟩R | ⟨rest⟩ Ai → ⟨activityi⟩Bi Ai | ⟨activityi⟩Bi
W[4,∞) → Ai for 1 ≤ i ≤m P→ W ⟨break⟩W
F → PLP S → RP[13,24]R | RF[30,38]R
L → ⟨lunch⟩ ⟨lunch⟩ ⟨lunch⟩ ⟨lunch⟩
Sa
w→
BPb
can be applied only if the sequence
the span of the literal C is between 1 and
5, or 7. The weight associated with this
unconstrained.
useful to model situations where a worker
changing activities or can have a lunch break
used to encode preferences or to model
employee doing some over-time.
minor modifications to the algorithm. On
production APa
Sa
w→
BPb
SbCPc
Sc , we add the
∈ Pc as well as j ∈ Sa, k ∈ Sb, and
12, 23] 24, 40]

19. Filtering
the
Grammar
Constraint
• Filtering
algorithm
by
Meinolf
Sellmann
in
2006,
Quimper
Walsh
2006
• Decomposi@on
by
Quimper
Walsh
2007
• Incremental
filtering
algorithm
by
Kadogliu
Sellmann
2010.
!
• A
boolean
representa@on
of
a
parsing
tree:
– A
leaf
is
true
if
its
corresponding
assignment
is
true
– If
a
AND
node
is
true,
both
its
children
must
be
true
– If
a
OR
node
is
true,
one
its
children
must
be
true
– if
a
node
is
true
(OR/AND)
one
of
its
parent
must
be
true
– The
root
is
true:
N(S,1,3)
=
1
!
• This
model
can
be
used
in
CP
or
SAT.
19

20. SAT
vs
CP,
Monoli@c
vs
Decomposi@ons
20
|A| # m GAC SAT SAT Mono Decomp
sol time bt opt sol time bt opt sol bt sol bt
1 2 4 26.0 2666 507215 p 26.0 1998 443546 p 26.75 28072 26.25 625683
1 3 6 37.25 1128199 36.75 1953562 37.0 34788 37.0 4771577
1 4 6 38.0 256 84999 p 38.0 287 91151 p - 15539 38.0 56488
1 5 5 24.0 153 67376 p 24.0 60 40008 p 24.0 40163 24.0 7914413
1 6 6 33.0 98 48638 p 33.0 70 40361 p - 11537 33.0 33405
1 7 8 49.5 236066 49.5 682715 49.0 27635 49.0 2663721
1 8 3 20.5 80 36348 p 20.5 44 25502 p 21.0 24343 20.5 635589
1 10 9 54.0 202699 54.25 507749 - 9365 - 519446
2 1 5 25.0 453 146234 25.0 301 103918 p 25.0 1180 25.0 3828461
2 2 10 58.75 313644 59.0 151076 58.0 14887 58.0 2116602
2 3 6 38.25 236850 38.25 214203 41.0 1419 41.0 214201
2 4 11 71.25 230777 69.75 239519 - 9983 - 774942
2 5 4 23.75 2945 644496 p 23.75 1876 522780 p 26.5 25573 26.25 117105
2 6 5 26.75 4831 777572 p 27.25 2162816 26.75 10681 26.75 1054531
2 8 5 31.5 244837 31.75 391858 32.0 218 31.5 3771831
2 9 3 19.0 2283 701474 p 19.0 1227 481395 p 19.25 20473 19.0 45516
2 10 8 55.0 372870 55.0 333520 - 9968 - 909857
Table 1. Benchmark problems solved by MiniSat+. GAC SAT: results from MiniSat+ with all
CNF clauses; SAT: results from MiniSat+ with all CNF clauses but clause (4); Mono: results
from a CP solver using the monolithic propagator; Decomp: results from a CP solver using the

21. Using
Lazy-­‐Clause
Genera@on
Table : Comparison of different methods on shift scheduling problems.
21
• ShiI
constraints
are
described
with
a
context-­‐free
grammar,
which
is
then
Experimental Results - Shift Scheduling
converted
into
an
MDD.
• Coverage
constraints
are
implemented
with
a
BDD
decomposi@on.
• Solved
with
LCG
with
explana@ons
over
MDD
with
costs
Inst. decmdd mdd ev-mdd ev-mddI
time fails time fails time fails time fails
1,2,4 14.51 39700 3.49 21888 0.20 607 0.17 635
1,3,6 11.25 40675 19.00 76348 0.87 4045 0.91 4156
1,4,6 2.62 7582 0.69 3518 0.11 350 0.27 1077
1,5,5 0.41 1585 0.52 3955 0.07 239 0.06 238
1,6,6 0.40 1412 0.21 1161 0.08 249 0.11 413
1,7,8 4.03 13149 2.43 12046 0.73 3838 0.83 4279
1,8,3 0.85 5002 0.39 3606 State-­‐0.06 of-­‐the-­‐
219 0.07 262
1,10,9 17.55 44222 19.77 68688 1.23 5046 1.31 7419
2,1,5 0.14 691 0.24 1490 0.02 art
78 0.01 45
2,2,10 — — 131.29 286747 43.62 99583 49.05 100958
2,3,6 188.77 187760 144.99 289568 2.39 for
1-­‐2
6443 5.94 13695
2,4,11 — — 391.59 918438 42.38 111567 2,5,4 25.85 59635 12.18 50340 0.65 ac:vi:es
92.89 220568
1545 0.48 1541
2,6,5 83.78 104911 30.27 80046 6.18 12100 7.63 16074
2,8,5 90.28 153331 34.69 110917 4.99 15507 10.02 26565
2,9,3 6.10 20472 9.17 42105 0.86 1898 0.47 1593
2,10,8 349.88 303227 95.61 168720 8.85 26331 17.22 37356
Total — — 896.53 2139581 113.29 289645 187.44 436874
Mean — — 52.74 125857.71 6.66 17037.94 11.03 25698.47
Geom. — — 7.92 31465.82 0.86 2667.65 1.03 3428.35

22. 22
!
Solving
SSP
with
Regular
and
Context-­‐free
Grammar
in
Mixed
Integer
Programming
• Côté, M. C., Gendron, B., Quimper, C. G., Rousseau, L. M. (2011).
Formal languages for integer programming modeling of shift scheduling
problems. Constraints, 16(1), 54-76.
!
• Côté, M. C., Gendron, B., Rousseau, L. M. (2011). Grammar-based
integer programming models for multiactivity shift scheduling. Management
Science, 57(1), 151-163.

23. Compact
assignment
formula@on

24. A
Regular
language
for
our
example
Replacing
sequencing
rules
(4
to
8)

25. Deriving
a
network
flow
problem

26. Building
a
network
flow
problem
Network
flow
formula@on
Linking
constraints
Derived
automa:cally
from
the
automaton

27. Linearizing
a
context-­‐free
grammar
• The
model
should
find
a
tree
in
the
Γ
graph
• The
lineariza@on
of
the
CFG
constraint
is
simply:
27
vc(u)
av =
vp(u)
av ⇤u ⇥ Nor
vc(R)
av = 1
av 0, ⇤v ⇥ Nand

28. 2 162,9396 31,67 3040 4285 163,9210 66,26
3 168,8223 41,62 3040 4285 170,5663 46,74
4 131,6560 42,89 2459 3408 133,1414 98,00
5 144,4182 52,52 2143 2956 145,2850 21,83
6 133,0766 57,56 2143 2956 135,1286 1,56
7 149,2739 44,03 2456 3405 REGULAR
150,7675 53,82
8 147,2002 47,63 3040 4285 148,0467 263,43
9 142,4836 27,46 1957 2614 182,4833 12,91
10 146,2410 37,19 2458 3408 147,6853 19,28
Results
on
our
examples
28
Results for the classical MIP model
ASSIGNMENT
MIP
Time |C| |V | MIP Value MIP Time
36,53 29871 4040 172,6670 2142,17
104,73 56743 5104
89,06 56743 5104
32,95 45983 4704 152,2240 3610,00
63,61 40447 4360 171,9930 3610,01
36,65 40447 4360 137,5180 3616,57
38,26 45887 4608
90,49 56743 5104
27,15 36175 4156 182,5370 3607,76
36,89 45983 4704 149,1810 3611,72
Table 4: Results for the complete MIP grammar model
GRAMMAR
LP MIP
Id LP Value LP Time |C| |V | MIP Value MIP Time
1 3612,35 2727 6356 172,6665 7,42
2 3610,68 35419 190480
3 3615,64 35419 190480
4 3614,92 19163 84672 133,0630 1850,38
5 3609,11 10387 41464 145,8830 322,57
6 3618,18 10387 41464 134,8178 130,21
7 3618,83 19163 84672 151,2082 1662,75
8 3611,39 35419 190480
9 3614,71 5503 21580 182,5370 1015,10
10 3610,10 19163 84672 147,0870 1313,28

29. Removing
Symmetries
• In
the
original
model
of
Dantzig
the
variables
actually
were
defined
to
capture
the
number
of
employees
working
on
each
shiI
!
• Which
removes
symmetries
in
the
case
where
employees
are
iden:cal
!
• Can
we
ask
the
same
ques@on
using
formal
language
based
MIP
?
29
NO YES

30. Implicit
Grammar
in
MIP
• Every
node
in
the
three
now
generates
general
integers:
– av
– NS
vc(u)
=
number
=
total
av =
of
@me
node
number
of
employees
needed
vp(u)
and
is
selected
by
an
employee
av ⇤u ⇥ Nor
vc(R)
av = 1
av Ns
0, ⇤v ⇥ Nand
This model was shown to have the Linear
Relaxation as mono-activity models of !
!
• Set Covering formulation of Dantzig!
• Implicit models of Aykin, Bechtold
Jacob, and Rekik.!
•

31. Results
on
the
full
mul@-­‐ac@vity
problem
• From
1
to
10
ac@vi@es
• number
of
shiI
increases
by
a
factor
of
10,000
• number
of
contraints
increases
by
a
factor
of
10.
– Grammar
based
model:
• number
of
variables
is
less
than
doubled
• number
of
constraint
increases
by
about
50%
• All
Cˆot´e, Gendron, and Rousseau: Grammar-Based Integer Programing Models for Multi-Activity Shift Scheduling
problem
solved
to
op:mality
in
less
than
10
minutes
Article submitted to Management Science; manuscript no. MS-0001-1922.65 Table 4 Multi-activity problems with the Implicit Grammar model
31
NbAct NbShifts |C| |V | Time Nopt(10)
2 13404928 70143 18202 423,90 9
3 67752783 73529 20144 337,14 9
4 214010944 76915 22092 212,07 9
5 522350575 80301 24046 182,02 10
6 1082991744 83687 26002 179,51 10
7 2006203423 87073 27970 317,31 10
8 3422303488 90459 29899 274,66 10
9 5481658719 93845 31821 296,37 10
10 8354684800 97231 33799 518,02 10

32. !
An
Hybrid
MIP
and
CP
Approach
32
• Salvagnin, D., Walsh, T. (2012). A hybrid MIP/CP approach for multi-activity
shift scheduling. In Principles and Practice of Constraint Programming (pp.
633-646). Springer Berlin Heidelberg.

33. ShiI
Scheduling
Problems:
Personalized
Cases
• Par@culari@es:
– List
of
available
employees
– Each
employee
has
individual
caracteris@cs
• Availability
within
the
planning
horizon
• Mul@-­‐ac@vity
case:
skills
to
perform
only
a
subset
of
the
ac@vi@es
•
Problem:
• Assign
one
shiI
to
each
employee;
• while
taking
into
account
the
constraints
on
the
composi@on
of
shiIs;
• minimizing
the
overall
cost.
!
• No
longer
possible
to
use
implicit
models…
at
least
not
solely
!
33

34. Orbital
Shrinking
• Given
a
model
contains
symmetries,
all
variables
which
are
symmetric
can
be
replace
by
a
single
variable
which
is
equal
to
their
sum.
!
!
!
!
!
!
!
!
!
• Thus
if
X1,
X2,
X3
are
totally
equal
(the
represent
iden@cal
employee),
we
replace
them
by
a
single
variable
34
variable orbits
constraint orbits
take averages!
within orbits
Y 3
1 =
X
i:1..3
Xi

35. Orbital
Shrinking
• This
trick
only
works
for
convex
op@miza@on
!
(not
MIP
in
general)
• Grammars
and
Regulars
allow
to
define
a
convex
space
!
!
!
!
!
!
!
!
• In
that
sense,
the
Implicit
Grammar
Model
was
a
case
of
Orbital
Shrinking.
!
• Now
if
the
employees
are
not
all
iden@cal,
can
we
s@ll
use
this
trick
?!?
35
Exact Reformulation
convex optimization
assignment
a
a
b b b
a a a
a a
regular/grammar constraint only

36. Hybrid An
Hybrid
MIP/MIP/CP
CP approach
decomposition
36
MIP solver on the
shrunken model
CP solver to check if a shrunken solution can be disaggregated
into a feasible solution of the original problem
inner node
pruned by bound
infeasible integer
solution
feasible integer
solution

37. Multi-phase approach
A
Mutliphase
Hybrid
MIP/CP
Approach
37
1.
solve aggregated model
and collect solutions
2.
apply MIP heuristic
to collected solutions
3.
final run with
real hybrid decomposition
✓ no callbacks!
✓ dual reductions on!
✓ very fast
could stop here!
if gap is small!
✓ callbacks!
✓ CP check!
✓ dual reductions off
Implicit
Grammar
MIP
formula@on
LNS
+
Grammar
DP
Integrated
CP/MIP
Branch
and
Bound

38. Experimental
Results
38
10
8
6
4
2
0
# solved
1 2 3 4 5 6 7 8 9 10
cpx-reg hybrid grammar
average time
10000
1000
100
10
1
1 2 3 4 5 6 7 8 9 10
cpx-reg hybrid grammar
gap
100.0%
10.0%
1.0%
0.1%
1 2 3 4 5 6 7 8 9 10
cpx-reg hybrid

39. !
Dynamic
Programming
1
Large
Neighborhood
Searh
39
• Quimper, C. G., Rousseau, L. M. (2010). A large neighbourhood search
approach to the multi-activity shift scheduling problem. Journal of Heuristics,
16(3), 373-392.

40. A
Simple
Problem
• Lets
look
at
a
smaller
example:
– There
are
no
more
than
two
consecu@ve
work
period.
– The
day
starts
and
finishes
with
a
work
period.
– Overstaffing
and
understaffing
costs
1
unit
40

41. Local
Search
41
1
0
1
0
0
1
1
0
Marginal costs

42. Best
replacement
shiI
using
Regular
42
Schedule
regula@ons Schedule
length
1
0
1
1
1
1
0 1
Marginal costs

43. Best
replacement
shiI
using
Grammar
OS
14
OX
13
OX
OW
12 5
6
5
6
12 OW
1
2 1
5 32
3
2 5 3
12 AWWW,1
12 AWWW,1
1 1 0 1 1
11 OB
21 OW
OW
2 0 21 3 OB
OW
OW
31 31 2 41
1
Ow
1 1 1
2 0 3 0 2 1
1 1 Ow
21 0 0 1
11 Ob
31 AWw,1
31 Ow
21 Ob
31 Ow
41
ASXW,1
14 ASXW,2
14
AXWB,1
13
AXWB,1
32
AWw,1
11 ABb,1
21 AWw,1
21 ABb,1
31 AWw,1
41
2
0
0 3 2 1
5
w w b w
43
Marginal costs
1
0 0
0
1
2
2
1
1
1
2
1
3

44. Local
Search
44
1
0
0 1

45. A
simple
large
neighborhood
search
• Generate
a
random
solu@on
• Repeat
– Select
an
employee
and
remove
its
schedule
– Compute
the
over-­‐costs
and
under-­‐costs
for
every
ac@vity
– Find
the
best
replacing
solu@on
by:
• Solving
shortest
path
in
the
automaton
layered
graph
(DAG)
• Finding
the
lightest
tree
in
an
AND/OR
Graph
– Insert
the
new
schedule
in
the
solu@on
• Un@l
a
local
minimum
is
reached
• Restart
45

46. Experimental
Results
1
ac@vity,
compared
with
MIP
46

47. Experimental
Results
1
to
10
ac@vi@es
47
Ind
runs
Ind
runs
1
min
runs
1
min
runs
Times
in
milliseconds
to
construct
one
shiI

48. 48
!
Dynamic
Programming
2
Column
Genera@on
• Demassey, S., Pesant, G., Rousseau, L. M. (2006). A cost-regular based
hybrid column generation approach. Constraints, 11(4), 315-333.
!
• Côté, M. C., Gendron, B., Rousseau, L. M. (2013). Grammar-based
column generation for personalized multi-activity shift scheduling. INFORMS
Journal on Computing, 25(3), 461-474.

49. Column
Genera@on
Framework
49
Select
among
a
subset
of
possible
shiIs,
those
that
together
fit
best
the
demand
(using
a
LP
set
covering
formula@on)
Find
the
shiIs
that
we
haven’t
seen
that
would
improve
the
current
solute
and so on…
NEW
SHIFTS
DUAL
VALUES
How ?

50. Grammar-­‐based
Column
Genera@on
Set
covering
model:
Restricted
master
problem:
Set
of
periods
in
the
planning
horizon
Set
of
ac@vi@es
Set
of
employees
Set
of
allowed
shiIs
for
employee
e
Set
of
allowed
shiIs
for
employee
e
at
column
genera@on
itera@on
k
Integrality
constraints
relaxed
50
dual
values

51. 12 AWWW,1
12 AWWW,1
31 AWw,1
31 Ow
1
0
1
1
1
1
0 1
Grammar-­‐based
Column
Genera@on
approach:
Solving
the
subproblem
OS
14
OX
13
OX
OW
12 12 OW
32
11 OB
21 OW
OW
21 OB
31 OW
31 OW
41
Ow
21 Ow
11 Ob
21 Ob
31 Ow
41
ASXW,1
14 ASXW,2
14
AXWB,1
13
AXWB,1
32
AWw,1
11 ABb,1
21 AWw,1
21 ABb,1
31 AWw,1
41
2 0 3 0 2 1
51
dual
values

52. Computa@onal
Results
Tableau 6.2 Comparison with Demassey et al. (2006) approach
Comparison with existing IP formulations based on formal languages. Implicit Root Node Branch and Price
Grammar MIP
2+ activities
Table 5: Multi-activity problems with the Implicit Grammar model
NbA Time(s) NbIts NbCols NbS(0.01%) Time(s) NbS(1%)
Grammar-based CG
NbAct NbShif ts |C| |V | Time(s) Nopt(10)
2 13404928 18068 69893 409.07 10
3 67752783 19945 73152 205.38 9
4 214010944 21822 76417 300.47 10
5 522350575 23699 79688 146.16 10
6 1082991744 25576 82961 213.79 10
7 2006203423 27453 86246 230.88 10
8 3422303488 29330 89492 257.06 10
9 5481658719 31207 92731 289.08 10
10 8354684800 33084 96026 516.74 10
1 0.1 10 257 5 62 10
2 0.2 16 601 6 100 9
3 0.6 20 975 6 2074 8
4 1.1 25 1321 State-­‐5 of-­‐2096 the-­‐art
9
5 3.0 46 2321 for
0 personalized
2h 10
6 4.0 47 2457 9 915 10
7 6.6 47 3117 5 employees
2426 9
8 8.3 62 3459 7 2163 10
9 9.6 62 3626 5 1886 7
10 9.9 43 3405 6 3754 8
Demassey et al. (2006) CG
1 0.4 19 889 8 144 2 3.7 48 2340 8 394 3 2.0 52 2550 4 1592 4 12.5 103 5063 0 2h 5 6.2 86 4288 0 2h 6 13.8 130 6493 0 2h 7 18.4 137 6839 0 2h 8 25.4 155 7736 0 2h 9 25.9 155 7741 0 2h 10 42.0 179 8974 0 2h
State-­‐of-­‐the-­‐art
for
Iden:cal
Employess
52
Regular-based CG
approach, the multi-activity problems can easily be handled with a few more productions
than in the one-activity case, and results show that they can rapidly be solved on almost available instances. Note Used
that the week
growth form
in the number of constraints and variables increasing the number of work of
activities branching
is much slower than the increase in the number feasible shifts.

53. !
Retail
Scheduling
for
Profit…
when
the
objec@ve
is
non
linear
• Chapados, N., Joliveau, M., L’Ecuyer, P., Rousseau, L. M. Retail Store
Scheduling for Profit. European Journal of Operations Research, Forthcoming.

54. Mo@va@on
• How
useful
are
salespersons
in
retail…
54

55. Fundamental
Accoun@ng
Equa@on
Profit
Sales
x
Margin
Salaries
Goal: maximize this

56. Sales
Decomposi@on
per
Period
Sales Traffic Conversion
Rate Average
Basket
Nb of shoppers
entering the store
Fraction of shoppers
that become buyers Amount bought,
knowing that something
was bought

57. Salary
Decomposi@on
Salary Hours
Worked
Hourly
Wage
Commission

58. A
Constraint
Programming
model
• Given
the
sets
:of
@me
periods
T,
of
employee
E
and
ac@vi@es
A
• inputs
– Rn
t
the
revenues
generated
if
n
people
are
working
at
@me
t.
– Ct
a
the
payroll
cost
of
an
employee
working
at
@me
t
• variables
– xt
e
∈
A
specifies
the
ac@vity
performed
by
employee
e
at
@me
t.
– yt
a
∈
{1..|E|}
specifies
the
number
of
employees
performing
a
at
@me
t
Tue
Mon
Sun
09:00 09:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:00 16:30 17:00 17:30 18:00 18:30 19:00 19:30 20:00 20:30
Fig. 2. Average revenue increase by day-of-week and time-of-day across all stores covered case study.
e(
max
Ca
ya
t t s.t. Regular(, xX
tT
Rty
w
t
X
aA
all tT)) ⇤e ⇥ E GCC(x(all eE)
t , A, y(all aA)
t ) ⇤t ⇥ T ya
t ⇥ {0, . . . , |E|} ⇤a ⇥ A, t ⇥ T xet
⇥ A ⇤e ⇥ E, t ⇥ T The model is based on two sets of variables: xet
specifies the activity (among

59. MIP
model
• Given
the
sets
:of
@me
periods
T,
of
employee
E
and
ac@vi@es
A
• inputs
– Rn
t
the
revenues
generated
if
n
people
are
working
at
@me
t
– Ct
a
the
payroll
cost
of
an
employee
working
at
@me
t
• variables
– xa
et
∈
{0,1}
1
if
ac@vity
a
is
performed
by
employee
e
at
@me
t.
– ya
kt
∈
{0,1}
1
if
exactly
k
employees
are
performing
a
at
@me
t

60. Scheduling
over
a
day
Table 7: Estimated relative profit increase p and relative gap to the best upper bound, for the
solutions obtained by solving the proposed CP model after 30 seconds and the MIP and CP models after
300 seconds of CPU time.
60
CP after 30 sec. CP and MIP after 300 sec.
Instances description p (%) (%) p (%) (%)
Store #Days #Periods #Empl. CP CP MIP CP MIP CP
1 408 33.8 6 3.4 0.3 3.5 3.5 0.2 0.2
2 326 32.4 5 2.8 0.2 2.6 2.6 0.1 0.1
3 372 40.6 5 2.6 0.2 2.6 2.6 0.1 0.1
4 377 36.7 5 2.9 0.2 3.0 3.0 0.1 0.1
5 309 40.4 5 2.6 0.2 2.7 2.7 0.1 0.1
6 204 32.0 6 4.0 0.2 3.4 4.2 0.8 0.0
7 399 37.5 5 2.7 0.2 2.8 2.8 0.1 0.1
8 384 41.7 6 3.0 0.3 2.0 3.2 1.4 0.2
9 241 33.0 5 2.9 0.2 3.0 3.1 0.1 0.0
10 401 33.4 4 1.3 0.0 1.3 1.3 0.0 0.0
11 369 42.0 5 2.1 0.2 2.2 2.2 0.2 0.2
12 381 40.3 5 2.2 0.1 2.2 2.2 0.1 0.0
13 366 41.4 5 2.2 0.1 2.1 2.3 0.2 0.1
14 368 42.6 8 2.9 0.5 2.8 3.2 0.6 0.2
15 184 42.1 9 2.7 0.8 2.4 3.2 1.1 0.3
Average 339.3 38.0 5.6 2.7 0.2 2.6 2.8 0.3 0.1
and
yw
d,t ! 2 8d 2 D, t 2 Td, (51)

61. Scheduling
over
a
week
61
Table 8: Evolution of the relative gap for the solutions identified by CP when building schedules over
a week with two hours of CPU time, in comparison with the best upper bound obtained after 24 hours.
Instances description Relative gap bound (in %)
Store #Weeks #Periods #Empl. after 30 min after 60 min after 120 min
1 37 236 6 2.2 2.0 1.6
2 23 224 5 1.9 1.7 1.5
3 21 282 5 3.8 3.6 3.4
4 34 256 5 2.5 2.3 2.1
5 17 280 5 4.4 4.0 3.6
6 15 232 6 2.8 2.5 2.4
7 38 262 5 3.1 2.7 2.5
8 38 288 6 3.4 3.3 3.1
9 14 230 5 2.0 1.8 1.6
10 36 234 4 2.0 1.9 1.7
11 22 294 5 3.4 3.3 3.2
12 23 282 5 2.9 2.7 2.6
13 22 288 5 3.0 2.8 2.7
14 18 296 8 1.9 1.7 1.7
15 12 292 9 8.3 7.2 6.9
Average 24.7 265.1 5.6 3.2 2.9 2.7
generate less profit than the solutions of the full MIP while using the same amount of
CPU time to construct feasible schedules, we always used the full MIP (no PLA) in this
section. For many instances, the MIP failed to return a feasible solution after 30 seconds;
for this reason we do not report the MIP results for 30 seconds. The solutions returned

62. !
An
Applica@on
to
Hydro-­‐Québec
Call-­‐Center
Scheduling
62
• Pelleau M, Rousseau L-M, L’Écuyer P, Zegal W, Delorme L. (2014),
Scheduling of Agents from Forecasted Future Call Arrivals at Hydro-Québec’s
Call Centers, In Principles and Practice of Constraint Programming. Springer
Berlin Heidelberg.
I HOPE YOU HAVE SEEN IT ON TUESDAY !

63. Experimental
Results
63
Results
tool
Objective Value with HQ’s tool
b. Phone First
eDrseitloérdmeeMontréSaclh2e. dPuolilnytgecohf nAigqeunetsMontréal3. Institut deMraeych7e, r2c0h1e4d’Hyd1ro5-/Q1u7ébec)
10 %
20 %
30 %
Improvement in agent
staffing levels met

64. QUESTIONS
?
64